Optimal. Leaf size=550 \[ \frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{8 e^8 (d+e x)^8}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (3 A e (2 c d-b e)-B \left (7 c d^2-e (4 b d-a e)\right )\right )}{7 e^8 (d+e x)^7}+\frac {\left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{2 e^8 (d+e x)^6}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{5 e^8 (d+e x)^5}+\frac {B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^8 (d+e x)^4}+\frac {c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{e^8 (d+e x)^3}+\frac {c^2 (7 B c d-3 b B e-A c e)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)} \]
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Rubi [A]
time = 0.42, antiderivative size = 548, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {785}
\begin {gather*} \frac {A e (2 c d-b e) \left (-2 c e (5 b d-3 a e)+b^2 e^2+10 c^2 d^2\right )-B \left (3 c e^2 \left (a^2 e^2-8 a b d e+10 b^2 d^2\right )-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+35 c^3 d^4\right )}{5 e^8 (d+e x)^5}+\frac {c \left (A c e (2 c d-b e)-B \left (-c e (6 b d-a e)+b^2 e^2+7 c^2 d^2\right )\right )}{e^8 (d+e x)^3}+\frac {\left (a e^2-b d e+c d^2\right ) \left (B \left (-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)+7 c^2 d^3\right )-A e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )\right )}{2 e^8 (d+e x)^6}+\frac {B \left (-15 c^2 d e (3 b d-a e)+3 b c e^2 (5 b d-2 a e)-b^3 e^3+35 c^3 d^3\right )-3 A c e \left (-c e (5 b d-a e)+b^2 e^2+5 c^2 d^2\right )}{4 e^8 (d+e x)^4}-\frac {\left (a e^2-b d e+c d^2\right )^2 \left (-B e (4 b d-a e)-3 A e (2 c d-b e)+7 B c d^2\right )}{7 e^8 (d+e x)^7}+\frac {(B d-A e) \left (a e^2-b d e+c d^2\right )^3}{8 e^8 (d+e x)^8}+\frac {c^2 (-A c e-3 b B e+7 B c d)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 785
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^3}{(d+e x)^9} \, dx &=\int \left (\frac {(-B d+A e) \left (c d^2-b d e+a e^2\right )^3}{e^7 (d+e x)^9}+\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{e^7 (d+e x)^8}+\frac {3 \left (c d^2-b d e+a e^2\right ) \left (-B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )+A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{e^7 (d+e x)^7}+\frac {-A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )+B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{e^7 (d+e x)^6}+\frac {-B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )+3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{e^7 (d+e x)^5}+\frac {3 c \left (-A c e (2 c d-b e)+B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{e^7 (d+e x)^4}+\frac {c^2 (-7 B c d+3 b B e+A c e)}{e^7 (d+e x)^3}+\frac {B c^3}{e^7 (d+e x)^2}\right ) \, dx\\ &=\frac {(B d-A e) \left (c d^2-b d e+a e^2\right )^3}{8 e^8 (d+e x)^8}-\frac {\left (c d^2-b d e+a e^2\right )^2 \left (7 B c d^2-B e (4 b d-a e)-3 A e (2 c d-b e)\right )}{7 e^8 (d+e x)^7}+\frac {\left (c d^2-b d e+a e^2\right ) \left (B \left (7 c^2 d^3-c d e (8 b d-3 a e)+b e^2 (2 b d-a e)\right )-A e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )\right )}{2 e^8 (d+e x)^6}+\frac {A e (2 c d-b e) \left (10 c^2 d^2+b^2 e^2-2 c e (5 b d-3 a e)\right )-B \left (35 c^3 d^4-b^2 e^3 (4 b d-3 a e)-30 c^2 d^2 e (2 b d-a e)+3 c e^2 \left (10 b^2 d^2-8 a b d e+a^2 e^2\right )\right )}{5 e^8 (d+e x)^5}+\frac {B \left (35 c^3 d^3-b^3 e^3+3 b c e^2 (5 b d-2 a e)-15 c^2 d e (3 b d-a e)\right )-3 A c e \left (5 c^2 d^2+b^2 e^2-c e (5 b d-a e)\right )}{4 e^8 (d+e x)^4}+\frac {c \left (A c e (2 c d-b e)-B \left (7 c^2 d^2+b^2 e^2-c e (6 b d-a e)\right )\right )}{e^8 (d+e x)^3}+\frac {c^2 (7 B c d-3 b B e-A c e)}{2 e^8 (d+e x)^2}-\frac {B c^3}{e^8 (d+e x)}\\ \end {align*}
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Mathematica [A]
time = 0.34, size = 847, normalized size = 1.54 \begin {gather*} -\frac {A e \left (5 c^3 \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )+e^3 \left (35 a^3 e^3+15 a^2 b e^2 (d+8 e x)+5 a b^2 e \left (d^2+8 d e x+28 e^2 x^2\right )+b^3 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+c e^2 \left (5 a^2 e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+6 a b e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+3 b^2 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+c^2 e \left (3 a e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )\right )+B \left (35 c^3 \left (d^7+8 d^6 e x+28 d^5 e^2 x^2+56 d^4 e^3 x^3+70 d^3 e^4 x^4+56 d^2 e^5 x^5+28 d e^6 x^6+8 e^7 x^7\right )+e^3 \left (5 a^3 e^3 (d+8 e x)+5 a^2 b e^2 \left (d^2+8 d e x+28 e^2 x^2\right )+3 a b^2 e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+b^3 \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )+c e^2 \left (3 a^2 e^2 \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+6 a b e \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )+5 b^2 \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )\right )+5 c^2 e \left (a e \left (d^5+8 d^4 e x+28 d^3 e^2 x^2+56 d^2 e^3 x^3+70 d e^4 x^4+56 e^5 x^5\right )+3 b \left (d^6+8 d^5 e x+28 d^4 e^2 x^2+56 d^3 e^3 x^3+70 d^2 e^4 x^4+56 d e^5 x^5+28 e^6 x^6\right )\right )\right )}{280 e^8 (d+e x)^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 1067, normalized size = 1.94 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 980, normalized size = 1.78 \begin {gather*} -\frac {280 \, B c^{3} x^{7} e^{7} + 35 \, B c^{3} d^{7} + 5 \, {\left (3 \, B b c^{2} e + A c^{3} e\right )} d^{6} + 140 \, {\left (7 \, B c^{3} d e^{6} + 3 \, B b c^{2} e^{7} + A c^{3} e^{7}\right )} x^{6} + 5 \, {\left (B b^{2} c e^{2} + {\left (B a e^{2} + A b e^{2}\right )} c^{2}\right )} d^{5} + 280 \, {\left (7 \, B c^{3} d^{2} e^{5} + B b^{2} c e^{7} + {\left (B a e^{7} + A b e^{7}\right )} c^{2} + {\left (3 \, B b c^{2} e^{6} + A c^{3} e^{6}\right )} d\right )} x^{5} + {\left (B b^{3} e^{3} + 3 \, A a c^{2} e^{3} + 3 \, {\left (2 \, B a b e^{3} + A b^{2} e^{3}\right )} c\right )} d^{4} + 70 \, {\left (35 \, B c^{3} d^{3} e^{4} + B b^{3} e^{7} + 3 \, A a c^{2} e^{7} + 5 \, {\left (3 \, B b c^{2} e^{5} + A c^{3} e^{5}\right )} d^{2} + 3 \, {\left (2 \, B a b e^{7} + A b^{2} e^{7}\right )} c + 5 \, {\left (B b^{2} c e^{6} + {\left (B a e^{6} + A b e^{6}\right )} c^{2}\right )} d\right )} x^{4} + 35 \, A a^{3} e^{7} + {\left (3 \, B a b^{2} e^{4} + A b^{3} e^{4} + 3 \, {\left (B a^{2} e^{4} + 2 \, A a b e^{4}\right )} c\right )} d^{3} + 56 \, {\left (35 \, B c^{3} d^{4} e^{3} + 3 \, B a b^{2} e^{7} + A b^{3} e^{7} + 5 \, {\left (3 \, B b c^{2} e^{4} + A c^{3} e^{4}\right )} d^{3} + 5 \, {\left (B b^{2} c e^{5} + {\left (B a e^{5} + A b e^{5}\right )} c^{2}\right )} d^{2} + 3 \, {\left (B a^{2} e^{7} + 2 \, A a b e^{7}\right )} c + {\left (B b^{3} e^{6} + 3 \, A a c^{2} e^{6} + 3 \, {\left (2 \, B a b e^{6} + A b^{2} e^{6}\right )} c\right )} d\right )} x^{3} + 5 \, {\left (B a^{2} b e^{5} + A a b^{2} e^{5} + A a^{2} c e^{5}\right )} d^{2} + 28 \, {\left (35 \, B c^{3} d^{5} e^{2} + 5 \, {\left (3 \, B b c^{2} e^{3} + A c^{3} e^{3}\right )} d^{4} + 5 \, B a^{2} b e^{7} + 5 \, A a b^{2} e^{7} + 5 \, A a^{2} c e^{7} + 5 \, {\left (B b^{2} c e^{4} + {\left (B a e^{4} + A b e^{4}\right )} c^{2}\right )} d^{3} + {\left (B b^{3} e^{5} + 3 \, A a c^{2} e^{5} + 3 \, {\left (2 \, B a b e^{5} + A b^{2} e^{5}\right )} c\right )} d^{2} + {\left (3 \, B a b^{2} e^{6} + A b^{3} e^{6} + 3 \, {\left (B a^{2} e^{6} + 2 \, A a b e^{6}\right )} c\right )} d\right )} x^{2} + 5 \, {\left (B a^{3} e^{6} + 3 \, A a^{2} b e^{6}\right )} d + 8 \, {\left (35 \, B c^{3} d^{6} e + 5 \, {\left (3 \, B b c^{2} e^{2} + A c^{3} e^{2}\right )} d^{5} + 5 \, {\left (B b^{2} c e^{3} + {\left (B a e^{3} + A b e^{3}\right )} c^{2}\right )} d^{4} + 5 \, B a^{3} e^{7} + 15 \, A a^{2} b e^{7} + {\left (B b^{3} e^{4} + 3 \, A a c^{2} e^{4} + 3 \, {\left (2 \, B a b e^{4} + A b^{2} e^{4}\right )} c\right )} d^{3} + {\left (3 \, B a b^{2} e^{5} + A b^{3} e^{5} + 3 \, {\left (B a^{2} e^{5} + 2 \, A a b e^{5}\right )} c\right )} d^{2} + 5 \, {\left (B a^{2} b e^{6} + A a b^{2} e^{6} + A a^{2} c e^{6}\right )} d\right )} x}{280 \, {\left (x^{8} e^{16} + 8 \, d x^{7} e^{15} + 28 \, d^{2} x^{6} e^{14} + 56 \, d^{3} x^{5} e^{13} + 70 \, d^{4} x^{4} e^{12} + 56 \, d^{5} x^{3} e^{11} + 28 \, d^{6} x^{2} e^{10} + 8 \, d^{7} x e^{9} + d^{8} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.35, size = 883, normalized size = 1.61 \begin {gather*} -\frac {35 \, B c^{3} d^{7} + {\left (280 \, B c^{3} x^{7} + 140 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{6} + 280 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} x^{5} + 70 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} x^{4} + 35 \, A a^{3} + 56 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} x^{3} + 140 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} x^{2} + 40 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )} e^{7} + {\left (980 \, B c^{3} d x^{6} + 280 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d x^{5} + 350 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d x^{4} + 56 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d x^{3} + 28 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d x^{2} + 40 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d x + 5 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d\right )} e^{6} + {\left (1960 \, B c^{3} d^{2} x^{5} + 350 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{2} x^{4} + 280 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{2} x^{3} + 28 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{2} x^{2} + 8 \, {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{2} x + 5 \, {\left (B a^{2} b + A a b^{2} + A a^{2} c\right )} d^{2}\right )} e^{5} + {\left (2450 \, B c^{3} d^{3} x^{4} + 280 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{3} x^{3} + 140 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{3} x^{2} + 8 \, {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{3} x + {\left (3 \, B a b^{2} + A b^{3} + 3 \, {\left (B a^{2} + 2 \, A a b\right )} c\right )} d^{3}\right )} e^{4} + {\left (1960 \, B c^{3} d^{4} x^{3} + 140 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{4} x^{2} + 40 \, {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{4} x + {\left (B b^{3} + 3 \, A a c^{2} + 3 \, {\left (2 \, B a b + A b^{2}\right )} c\right )} d^{4}\right )} e^{3} + 5 \, {\left (196 \, B c^{3} d^{5} x^{2} + 8 \, {\left (3 \, B b c^{2} + A c^{3}\right )} d^{5} x + {\left (B b^{2} c + {\left (B a + A b\right )} c^{2}\right )} d^{5}\right )} e^{2} + 5 \, {\left (56 \, B c^{3} d^{6} x + {\left (3 \, B b c^{2} + A c^{3}\right )} d^{6}\right )} e}{280 \, {\left (x^{8} e^{16} + 8 \, d x^{7} e^{15} + 28 \, d^{2} x^{6} e^{14} + 56 \, d^{3} x^{5} e^{13} + 70 \, d^{4} x^{4} e^{12} + 56 \, d^{5} x^{3} e^{11} + 28 \, d^{6} x^{2} e^{10} + 8 \, d^{7} x e^{9} + d^{8} e^{8}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1127 vs.
\(2 (560) = 1120\).
time = 1.33, size = 1127, normalized size = 2.05 \begin {gather*} -\frac {{\left (280 \, B c^{3} x^{7} e^{7} + 980 \, B c^{3} d x^{6} e^{6} + 1960 \, B c^{3} d^{2} x^{5} e^{5} + 2450 \, B c^{3} d^{3} x^{4} e^{4} + 1960 \, B c^{3} d^{4} x^{3} e^{3} + 980 \, B c^{3} d^{5} x^{2} e^{2} + 280 \, B c^{3} d^{6} x e + 35 \, B c^{3} d^{7} + 420 \, B b c^{2} x^{6} e^{7} + 140 \, A c^{3} x^{6} e^{7} + 840 \, B b c^{2} d x^{5} e^{6} + 280 \, A c^{3} d x^{5} e^{6} + 1050 \, B b c^{2} d^{2} x^{4} e^{5} + 350 \, A c^{3} d^{2} x^{4} e^{5} + 840 \, B b c^{2} d^{3} x^{3} e^{4} + 280 \, A c^{3} d^{3} x^{3} e^{4} + 420 \, B b c^{2} d^{4} x^{2} e^{3} + 140 \, A c^{3} d^{4} x^{2} e^{3} + 120 \, B b c^{2} d^{5} x e^{2} + 40 \, A c^{3} d^{5} x e^{2} + 15 \, B b c^{2} d^{6} e + 5 \, A c^{3} d^{6} e + 280 \, B b^{2} c x^{5} e^{7} + 280 \, B a c^{2} x^{5} e^{7} + 280 \, A b c^{2} x^{5} e^{7} + 350 \, B b^{2} c d x^{4} e^{6} + 350 \, B a c^{2} d x^{4} e^{6} + 350 \, A b c^{2} d x^{4} e^{6} + 280 \, B b^{2} c d^{2} x^{3} e^{5} + 280 \, B a c^{2} d^{2} x^{3} e^{5} + 280 \, A b c^{2} d^{2} x^{3} e^{5} + 140 \, B b^{2} c d^{3} x^{2} e^{4} + 140 \, B a c^{2} d^{3} x^{2} e^{4} + 140 \, A b c^{2} d^{3} x^{2} e^{4} + 40 \, B b^{2} c d^{4} x e^{3} + 40 \, B a c^{2} d^{4} x e^{3} + 40 \, A b c^{2} d^{4} x e^{3} + 5 \, B b^{2} c d^{5} e^{2} + 5 \, B a c^{2} d^{5} e^{2} + 5 \, A b c^{2} d^{5} e^{2} + 70 \, B b^{3} x^{4} e^{7} + 420 \, B a b c x^{4} e^{7} + 210 \, A b^{2} c x^{4} e^{7} + 210 \, A a c^{2} x^{4} e^{7} + 56 \, B b^{3} d x^{3} e^{6} + 336 \, B a b c d x^{3} e^{6} + 168 \, A b^{2} c d x^{3} e^{6} + 168 \, A a c^{2} d x^{3} e^{6} + 28 \, B b^{3} d^{2} x^{2} e^{5} + 168 \, B a b c d^{2} x^{2} e^{5} + 84 \, A b^{2} c d^{2} x^{2} e^{5} + 84 \, A a c^{2} d^{2} x^{2} e^{5} + 8 \, B b^{3} d^{3} x e^{4} + 48 \, B a b c d^{3} x e^{4} + 24 \, A b^{2} c d^{3} x e^{4} + 24 \, A a c^{2} d^{3} x e^{4} + B b^{3} d^{4} e^{3} + 6 \, B a b c d^{4} e^{3} + 3 \, A b^{2} c d^{4} e^{3} + 3 \, A a c^{2} d^{4} e^{3} + 168 \, B a b^{2} x^{3} e^{7} + 56 \, A b^{3} x^{3} e^{7} + 168 \, B a^{2} c x^{3} e^{7} + 336 \, A a b c x^{3} e^{7} + 84 \, B a b^{2} d x^{2} e^{6} + 28 \, A b^{3} d x^{2} e^{6} + 84 \, B a^{2} c d x^{2} e^{6} + 168 \, A a b c d x^{2} e^{6} + 24 \, B a b^{2} d^{2} x e^{5} + 8 \, A b^{3} d^{2} x e^{5} + 24 \, B a^{2} c d^{2} x e^{5} + 48 \, A a b c d^{2} x e^{5} + 3 \, B a b^{2} d^{3} e^{4} + A b^{3} d^{3} e^{4} + 3 \, B a^{2} c d^{3} e^{4} + 6 \, A a b c d^{3} e^{4} + 140 \, B a^{2} b x^{2} e^{7} + 140 \, A a b^{2} x^{2} e^{7} + 140 \, A a^{2} c x^{2} e^{7} + 40 \, B a^{2} b d x e^{6} + 40 \, A a b^{2} d x e^{6} + 40 \, A a^{2} c d x e^{6} + 5 \, B a^{2} b d^{2} e^{5} + 5 \, A a b^{2} d^{2} e^{5} + 5 \, A a^{2} c d^{2} e^{5} + 40 \, B a^{3} x e^{7} + 120 \, A a^{2} b x e^{7} + 5 \, B a^{3} d e^{6} + 15 \, A a^{2} b d e^{6} + 35 \, A a^{3} e^{7}\right )} e^{\left (-8\right )}}{280 \, {\left (x e + d\right )}^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 1115, normalized size = 2.03 \begin {gather*} -\frac {\frac {5\,B\,a^3\,d\,e^6+35\,A\,a^3\,e^7+5\,B\,a^2\,b\,d^2\,e^5+15\,A\,a^2\,b\,d\,e^6+3\,B\,a^2\,c\,d^3\,e^4+5\,A\,a^2\,c\,d^2\,e^5+3\,B\,a\,b^2\,d^3\,e^4+5\,A\,a\,b^2\,d^2\,e^5+6\,B\,a\,b\,c\,d^4\,e^3+6\,A\,a\,b\,c\,d^3\,e^4+5\,B\,a\,c^2\,d^5\,e^2+3\,A\,a\,c^2\,d^4\,e^3+B\,b^3\,d^4\,e^3+A\,b^3\,d^3\,e^4+5\,B\,b^2\,c\,d^5\,e^2+3\,A\,b^2\,c\,d^4\,e^3+15\,B\,b\,c^2\,d^6\,e+5\,A\,b\,c^2\,d^5\,e^2+35\,B\,c^3\,d^7+5\,A\,c^3\,d^6\,e}{280\,e^8}+\frac {x^4\,\left (B\,b^3\,e^3+5\,B\,b^2\,c\,d\,e^2+3\,A\,b^2\,c\,e^3+15\,B\,b\,c^2\,d^2\,e+5\,A\,b\,c^2\,d\,e^2+6\,B\,a\,b\,c\,e^3+35\,B\,c^3\,d^3+5\,A\,c^3\,d^2\,e+5\,B\,a\,c^2\,d\,e^2+3\,A\,a\,c^2\,e^3\right )}{4\,e^4}+\frac {x\,\left (5\,B\,a^3\,e^6+5\,B\,a^2\,b\,d\,e^5+15\,A\,a^2\,b\,e^6+3\,B\,a^2\,c\,d^2\,e^4+5\,A\,a^2\,c\,d\,e^5+3\,B\,a\,b^2\,d^2\,e^4+5\,A\,a\,b^2\,d\,e^5+6\,B\,a\,b\,c\,d^3\,e^3+6\,A\,a\,b\,c\,d^2\,e^4+5\,B\,a\,c^2\,d^4\,e^2+3\,A\,a\,c^2\,d^3\,e^3+B\,b^3\,d^3\,e^3+A\,b^3\,d^2\,e^4+5\,B\,b^2\,c\,d^4\,e^2+3\,A\,b^2\,c\,d^3\,e^3+15\,B\,b\,c^2\,d^5\,e+5\,A\,b\,c^2\,d^4\,e^2+35\,B\,c^3\,d^6+5\,A\,c^3\,d^5\,e\right )}{35\,e^7}+\frac {x^2\,\left (5\,B\,a^2\,b\,e^5+3\,B\,a^2\,c\,d\,e^4+5\,A\,a^2\,c\,e^5+3\,B\,a\,b^2\,d\,e^4+5\,A\,a\,b^2\,e^5+6\,B\,a\,b\,c\,d^2\,e^3+6\,A\,a\,b\,c\,d\,e^4+5\,B\,a\,c^2\,d^3\,e^2+3\,A\,a\,c^2\,d^2\,e^3+B\,b^3\,d^2\,e^3+A\,b^3\,d\,e^4+5\,B\,b^2\,c\,d^3\,e^2+3\,A\,b^2\,c\,d^2\,e^3+15\,B\,b\,c^2\,d^4\,e+5\,A\,b\,c^2\,d^3\,e^2+35\,B\,c^3\,d^5+5\,A\,c^3\,d^4\,e\right )}{10\,e^6}+\frac {x^5\,\left (B\,b^2\,c\,e^2+3\,B\,b\,c^2\,d\,e+A\,b\,c^2\,e^2+7\,B\,c^3\,d^2+A\,c^3\,d\,e+B\,a\,c^2\,e^2\right )}{e^3}+\frac {x^3\,\left (3\,B\,a^2\,c\,e^4+3\,B\,a\,b^2\,e^4+6\,B\,a\,b\,c\,d\,e^3+6\,A\,a\,b\,c\,e^4+5\,B\,a\,c^2\,d^2\,e^2+3\,A\,a\,c^2\,d\,e^3+B\,b^3\,d\,e^3+A\,b^3\,e^4+5\,B\,b^2\,c\,d^2\,e^2+3\,A\,b^2\,c\,d\,e^3+15\,B\,b\,c^2\,d^3\,e+5\,A\,b\,c^2\,d^2\,e^2+35\,B\,c^3\,d^4+5\,A\,c^3\,d^3\,e\right )}{5\,e^5}+\frac {c^2\,x^6\,\left (A\,c\,e+3\,B\,b\,e+7\,B\,c\,d\right )}{2\,e^2}+\frac {B\,c^3\,x^7}{e}}{d^8+8\,d^7\,e\,x+28\,d^6\,e^2\,x^2+56\,d^5\,e^3\,x^3+70\,d^4\,e^4\,x^4+56\,d^3\,e^5\,x^5+28\,d^2\,e^6\,x^6+8\,d\,e^7\,x^7+e^8\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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